A conjecture about Dirichlet series.

[tpm]

if $$g(s)= \sum_{n=1}^{\infty} a(n) n^{-s}$$

Where g(s) has a single pole at s=1 with residue C, then my question/conjecture is if for s >0 (real part of s bigger than 0) we can write

$$g(s)= C(\frac{1}{s-1}+1)-s\int_{0}^{\infty}dx(Cx-A(x))x^{-s-1}$$

of course $$A(x)=\sum_{n \le x}a(n)$$

the question is if the series converge for s >1 with a pole there is a method to 'substract' this singularity (pole) at s=1 to give meaning for the series at any positive s.

I think that the 'Ramanujan resummation' may help to give the result:

$$\sum_{ n >1}^{[R]}a(n)n^{-s} = g(s)-C(s-1)^{-1}$$ valid even for s=1 or s>0 (??)

 

[DeadWolfe]

Jose, how's it been?